Base field \(\Q(\sqrt{161}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 40 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-40, -1, 1]))
gp: K = nfinit(Polrev([-40, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-40, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,1]),K([-69349916,10132526]),K([-294952791826,43094732226])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,1]),Polrev([-69349916,10132526]),Polrev([-294952791826,43094732226])], K);
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![-69349916,10132526],K![-294952791826,43094732226]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a-5)\) | = | \((-a+7)\cdot(-6a-35)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-73a+505)\) | = | \((-a+7)^{3}\cdot(-6a-35)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 5000 \) | = | \(2^{3}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{105610969788907510039}{5000} a + \frac{77152625593754088522}{625} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-\frac{2630557}{700} a + \frac{18002437}{700} : -\frac{38853985089}{49000} a + \frac{265927766249}{49000} : 1\right)$ |
| Height | \(8.1279074821021334252437415991626545168\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(581 a - \frac{15917}{4} : -291 a + \frac{15913}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 8.1279074821021334252437415991626545168 \) | ||
| Period: | \( 1.1484058674408355787658803064400728265 \) | ||
| Tamagawa product: | \( 6 \) = \(3\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.2068990663546852428366989988807683490 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+7)\) | \(2\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((-6a-35)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
10.3-g
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.